describe the five distinct groups of order 8SolutionDescript

describe the five distinct groups of order 8

Solution

Description of the five distinct groups of order 8 :

The five distinct groups of order 8 are namely cyclic group Z8 ,Direct product of Z4 and Z2 ,dihedral group D8, quaternion group ,elementary abelion group E8.

Cyclic group Z8 :

It has the second part of gap ID 1,sub group rank 1,Nilpotent class 1,derived length 1,frattini length 1,minimum size of generating set 1,prime base logorithm of exponent 3.

Direct product of Z4 and Z2 :

.It has the second part of gap ID 2,sub group rank 1,Nilpotent class 1,derived length 1,frattini length 3,minimum size of generating set 1,prime base logorithm of exponent 2.

dihedral group D8 :

It has the second part of gap ID 3,sub group rank 2,Nilpotent class 2,derived length 2,frattini length 2,minimum size of generating set 2,prime base logorithm of exponent 2.

quaternion group :

It has the second part of gap ID 4,sub group rank 2,Nilpotent class 2,derived length 2,frattini length 2,minimum size of generating set 2,prime base logorithm of exponent 2.

elementary abelion group E8 :

It has the second part of gap ID 5,sub group rank 3,Nilpotent class 1,derived length 1,frattini length 1,minimum size of generating set 3,prime base logorithm of exponent 1.